 least squares approximation

in statistics, a method for estimating the true value of some quantity based on a consideration of errors (error) in observations or measurements. In particular, the line (function) that minimizes the sum of the squared distances (deviations) from the line to each observation is used to approximate a relationship that is assumed to be linear. The method has also been generalized for use with nonlinear relationships.One of the first applications of the method of least squares was to settle a controversy involving the shape of the Earth. The English mathematician Isaac Newton (Newton, Sir Isaac) asserted in the Principia (1687) that the Earth has an oblate (grapefruit) shape due to its spin—causing the equatorial diameter to exceed the polar diameter by about 1 part in 230. In 1718 the director of the Paris Observatory, Jacques Cassini (Cassini, Jacques), asserted on the basis of his own measurements that the Earth has a prolate (lemon) shape.To settle the dispute, in 1736 the French Academy of Sciences (Sciences, Academy of) sent surveying expeditions to Ecuador and Lapland. However, distances cannot be measured perfectly, and the measurement errors at the time were large enough to create substantial uncertainty. Several methods were proposed for fitting a line through this data—that is, to obtain the function (line) that best fit the data relating the measured arc length to the latitude. It was generally agreed that the method ought to minimize deviations in the ydirection (the arc length), but many options were available, including minimizing the largest such deviation and minimizing the sum of their absolute sizes (as depicted in the figure—>). The measurements seemed to support Newton's theory, but the relatively large error estimates for the measurements left too much uncertainty for a definitive conclusion—although this was not immediately recognized. In fact, while Newton was essentially right, later observations showed that his prediction for excess equatorial diameter was about 30 percent too large.In 1805 the French mathematician AdrienMarie Legendre (Legendre, AdrienMarie) published the first known recommendation to use the line that minimizes the sum of the squares of these deviations—i.e., the modern least squares approximation. The German mathematician Carl Friedrich Gauss (Gauss, Carl Friedrich), who may have used the same method previously, contributed important computational and theoretical advances. The method of least squares is now widely used for fitting lines and curves to scatterplots (discrete sets of data).Richard Routledge
* * *
Universalium. 2010.
Look at other dictionaries:
leastsquares approximation — aproksimacija minimaliųjų kvadratų metodu statusas T sritis automatika atitikmenys: angl. least squares approximation vok. Approximation durch kleinste Quadrate, f rus. аппроксимация методом наименьших квадратов, f pranc. approximation des… … Automatikos terminų žodynas
Leastsquares spectral analysis — (LSSA) is a method of estimating a frequency spectrum, based on a least squares fit of sinusoids to data samples, similar to Fourier analysis. [cite book  title = Variable Stars As Essential Astrophysical Tools  author = Cafer Ibanoglu … … Wikipedia
Least squares — The method of least squares is a standard approach to the approximate solution of overdetermined systems, i.e., sets of equations in which there are more equations than unknowns. Least squares means that the overall solution minimizes the sum of… … Wikipedia
Least Squares — Die Methode der kleinsten Quadrate (bezeichnender auch: der kleinsten Fehlerquadrate; englisch: Least Squares Method) ist das mathematische Standardverfahren zur Ausgleichungsrechnung. Es ist eine Wolke aus Datenpunkten gegeben, die physikalische … Deutsch Wikipedia
Least Squares — A statistical method used to determine a line of best fit by minimizing the sum of squares created by a mathematical function. A square is determined by squaring the distance between a data point and the regression line. The least squares… … Investment dictionary
Nonlinear least squares — is the form of least squares analysis which is used to fit a set of m observations with a model that is non linear in n unknown parameters (m > n). It is used in some forms of non linear regression. The basis of the method is to… … Wikipedia
Linear least squares (mathematics) — This article is about the mathematics that underlie curve fitting using linear least squares. For statistical regression analysis using least squares, see linear regression. For linear regression on a single variable, see simple linear regression … Wikipedia
Ordinary least squares — This article is about the statistical properties of unweighted linear regression analysis. For more general regression analysis, see regression analysis. For linear regression on a single variable, see simple linear regression. For the… … Wikipedia
Moving least squares — is a method of reconstructing continuous functions from a set of unorganized point samples via the calculation of a weighted least squares measure biased towards the region around the point at which the reconstructed value is requested. In… … Wikipedia
Total least squares — The bivariate (Deming regression) case of Total Least Squares. The red lines show the error in both x and y. This is different from the traditional least squares method which measures error parallel to the y axis. The case shown, with deviations… … Wikipedia